The equation of state

  1. I've got some difficulties trying to understand the equation of state derived from Friedmann equations. I'd greatly appreciate it if someone walked me through this.

    Now if the equation of state is stated as:
    [tex]\Large \dot{\rho}+(3\rho +p)\frac{\dot{R}}{R}=0 \ \ |p=\omega \rho[/tex]

    Then (in the case of pressure being zero):
    [tex]\Rightarrow \rho \propto R^{-3} \Rightarrow \rho = \rho _0 (\frac{R_0}{R})^3[/tex]
    I suspect the latter to be correct as it's not a result of my own logic :biggrin:

    Now what I do not understand is the proportionality. If the equation of state is integrated I get something like this (set p=0):
    [tex]\Large \dot{\rho}+(3\rho)\frac{\dot{R}}{R}=0
    \Rightarrow \dot{\rho}=-3\rho\frac{\dot{R}}{R} \Rightarrow
    \frac{1}{\rho}\dot{\rho}=-3\frac{\dot{R}}{R} \Rightarrow
    \frac{1}{\rho}\frac{d\rho}{dt}=-3\frac{1}{R}\frac{dR}{dt}\ \|\cdot dt[/tex]
    \int _{\rho _0}^\rho \frac{1}{\rho}d\rho}=-3\int _{R_0}^R \frac{1}{R}dR \Rightarrow
    ln(\rho)-ln(\rho _0)=-3(ln(R)-ln(R_0)) \Rightarrow
    ln \frac{\rho}{\rho _0}=-3ln\frac{R}{R_0}\Rightarrow
    \frac{\rho}{\rho _0}=e^-3\frac{R}{R_0} \Rightarrow
    \rho=e^-3\frac{R}{R_0}\rho _0[/tex]

    Now is there some part to the theory that causes the equation to flip so that
    [tex]\rho =e^-3\frac{R}{R_0}\rho _0 \Rightarrow \rho = \rho _0 (\frac{R_0}{R})^3[/tex]
    or don't I just get the mathematics right?

    Or have I done something wrong right in the beginning deriving the equation of state?

    Of course, how could I not see it... forgetting that a log x = log xa.

    Kurdt already told me that here, but then his message disappeared.

    Well thanks to Kurdt anyway!
    Last edited: Dec 10, 2006
  2. jcsd
  3. Kurdt

    Kurdt 4,851
    Staff Emeritus
    Science Advisor
    Gold Member

    Simply the laws of logarithms:

    [tex] a\log(b) = \log(b^a)[/tex]

    and noting

    [tex] (\frac{a}{b})^{-3} = (\frac{b}{a})^3 [/tex]

    There is also no exponential function involved since:

    [tex] e^{(\ln x)}=x[/tex]

    EDIT: I'd deleted the original post because I thought I'd misread your original post but I had not.
    Last edited: Dec 10, 2006
  4. hellfire

    hellfire 1,047
    Science Advisor

    Just as a side note, the correct equation is:

    [tex]\Large \dot{\rho}+3\frac{\dot{R}}{R}(\rho +p)=0[/tex]

    It follows from [itex]\nabla_{\mu}T^{\mu}_0 = 0[/itex].
Know someone interested in this topic? Share this thead via email, Google+, Twitter, or Facebook

Have something to add?